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Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.

Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function.

A function ${\displaystyle f}$ is normally thought of as acting on the points in the function domain by "sending" a point ${\displaystyle x}$ in the domain to the point ${\displaystyle f(x).}$ Instead of acting on points, distribution theory reinterprets functions such as ${\displaystyle f}$ as acting on test functions in a certain way. In applications to physics and engineering, test functions are usually infinitely differentiable complex-valued (or real-valued) functions with compact support that are defined on some given non-empty open subset ${\displaystyle U\subseteq \mathbb {R} ^{n}}$. (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by ${\displaystyle C_{c}^{\infty }(U)}$ or ${\displaystyle {\mathcal {D}}(U).}$

Most commonly encountered functions, including all continuous maps ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ if using ${\displaystyle U:=\mathbb {R} ,}$ can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that such a function ${\displaystyle f}$ "acts on" a test function ${\displaystyle \psi \in {\mathcal {D}}(\mathbb {R} )}$ by "sending" it to the number ${\textstyle \int _{\mathbb {R} }f\,\psi \,dx,}$ which is often denoted by ${\displaystyle D_{f}(\psi ).}$ This new action ${\textstyle \psi \mapsto D_{f}(\psi )}$ of ${\displaystyle f}$ defines a scalar-valued map ${\displaystyle D_{f}:{\mathcal {D}}(\mathbb {R} )\to \mathbb {C} ,}$ whose domain is the space of test functions ${\displaystyle {\mathcal {D}}(\mathbb {R} ).}$ This functional ${\displaystyle D_{f}}$ turns out to have the two defining properties of what is known as a distribution on ${\displaystyle U=\mathbb {R} }$: it is linear, and it is also continuous when ${\displaystyle {\mathcal {D}}(\mathbb {R} )}$ is given a certain topology called the canonical LF topology. The action (the integration ${\textstyle \psi \mapsto \int _{\mathbb {R} }f\,\psi \,dx}$) of this distribution ${\displaystyle D_{f}}$ on a test function ${\displaystyle \psi }$ can be interpreted as a weighted average of the distribution on the support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like ${\displaystyle D_{f}}$ that arise from functions in this way are prototypical examples of distributions, but there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions ${\textstyle \psi \mapsto \int _{U}\psi d\mu }$ against certain measures ${\displaystyle \mu }$ on ${\displaystyle U.}$ Nonetheless, it is still always possible to reduce any arbitrary distribution down to a simpler family of related distributions that do arise via such actions of integration.

More generally, a distribution on ${\displaystyle U}$ is by definition a linear functional on ${\displaystyle C_{c}^{\infty }(U)}$ that is continuous when ${\displaystyle C_{c}^{\infty }(U)}$ is given a topology called the canonical LF topology. This leads to the space of (all) distributions on ${\displaystyle U}$, usually denoted by ${\displaystyle {\mathcal {D}}'(U)}$ (note the prime), which by definition is the space of all distributions on ${\displaystyle U}$ (that is, it is the continuous dual space of ${\displaystyle C_{c}^{\infty }(U)}$); it is these distributions that are the main focus of this article.

Definitions of the appropriate topologies on spaces of test functions and distributions are given in the article on spaces of test functions and distributions. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.

History

The practical use of distributions can be traced back to the use of Green's functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to Kolmogorov & Fomin (1957), generalized functions originated in the work of Sergei Sobolev (1936) on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. Gårding (1997) comments that although the ideas in the transformative book by Schwartz (1951) were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference. A detailed history of the theory of distributions was given by Lützen (1982).

Notation

The following notation will be used throughout this article:

• ${\displaystyle n}$ is a fixed positive integer and ${\displaystyle U}$ is a fixed non-empty open subset of Euclidean space ${\displaystyle \mathbb {R} ^{n}.}$
• ${\displaystyle \mathbb {N} _{0}=\{0,1,2,\ldots \}}$ denotes the natural numbers.
• ${\displaystyle k}$ will denote a non-negative integer or ${\displaystyle \infty .}$
• If ${\displaystyle f}$ is a function then ${\displaystyle \operatorname {Dom} (f)}$ will denote its domain and the support of ${\displaystyle f,}$ denoted by ${\displaystyle \operatorname {supp} (f),}$ is defined to be the closure of the set ${\displaystyle \{x\in \operatorname {Dom} (f):f(x)\neq 0\}}$ in ${\displaystyle \operatorname {Dom} (f).}$
• For two functions ${\displaystyle f,g:U\to \mathbb {C} ,}$ the following notation defines a canonical pairing: $${\displaystyle \langle f,g\rangle :=\int _{U}f(x)g(x)\,dx.}$$
• A multi-index of size ${\displaystyle n}$ is an element in ${\displaystyle \mathbb {N} ^{n}}$ (given that ${\displaystyle n}$ is fixed, if the size of multi-indices is omitted then the size should be assumed to be ${\displaystyle n}$). The length of a multi-index ${\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {N} ^{n}}$ is defined as ${\displaystyle \alpha _{1}+\cdots +\alpha _{n}}$ and denoted by ${\displaystyle |\alpha |.}$ Multi-indices are particularly useful when dealing with functions of several variables, in particular, we introduce the following notations for a given multi-index ${\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {N} ^{n}}$: $${\displaystyle {\begin{aligned}x^{\alpha }&=x_{1}^{\alpha _{1}}\cdots x_{n}^{\alpha _{n}}\\\partial ^{\alpha }&={\frac {\partial ^{|\alpha |}}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}\end{aligned}}}$$ We also introduce a partial order of all multi-indices by ${\displaystyle \beta \geq \alpha }$ if and only if ${\displaystyle \beta _{i}\geq \alpha _{i}}$ for all ${\displaystyle 1\leq i\leq n.}$ When ${\displaystyle \beta \geq \alpha }$ we define their multi-index binomial coefficient as: $${\displaystyle {\binom {\beta }{\alpha }}:={\binom {\beta _{1}}{\alpha _{1}}}\cdots {\binom {\beta _{n}}{\alpha _{n}}}.}$$

Definitions of test functions and distributions

In this section, some basic notions and definitions needed to define real-valued distributions on U are introduced. Further discussion of the topologies on the spaces of test functions and distributions is given in the article on spaces of test functions and distributions.

1. Let ${\displaystyle k\in \{0,1,2,\ldots ,\infty \}.}$
2. Let ${\displaystyle C^{k}(U)}$ denote the vector space of all k-times continuously differentiable real or complex-valued functions on U.
3. For any compact subset ${\displaystyle K\subseteq U,}$ let ${\displaystyle C^{k}(K)}$ and ${\displaystyle C^{k}(K;U)}$ both denote the vector space of all those functions ${\displaystyle f\in C^{k}(U)}$ such that ${\displaystyle \operatorname {supp} (f)\subseteq K.}$
   • If ${\displaystyle f\in C^{k}(K)}$ then the domain of ${\displaystyle f}$ is U and not K. So although ${\displaystyle C^{k}(K)}$ depends on both K and U, only K is typically indicated. The justification for this common practice is detailed below. The notation ${\displaystyle C^{k}(K;U)}$ will only be used when the notation ${\displaystyle C^{k}(K)}$ risks being ambiguous.
   • Every ${\displaystyle C^{k}(K)}$ contains the constant 0 map, even if ${\displaystyle K=\varnothing .}$
4. Let ${\displaystyle C_{c}^{k}(U)}$ denote the set of all ${\displaystyle f\in C^{k}(U)}$ such that ${\displaystyle f\in C^{k}(K)}$ for some compact subset K of U.
   • Equivalently, ${\displaystyle C_{c}^{k}(U)}$ is the set of all ${\displaystyle f\in C^{k}(U)}$ such that ${\displaystyle f}$ has compact support.
   • ${\displaystyle C_{c}^{k}(U)}$ is equal to the union of all ${\displaystyle C^{k}(K)}$ as ${\displaystyle K\subseteq U}$ ranges over all compact subsets of ${\displaystyle U.}$
   • If ${\displaystyle f}$ is a real-valued function on ${\displaystyle U}$, then ${\displaystyle f}$ is an element of ${\displaystyle C_{c}^{k}(U)}$ if and only if ${\displaystyle f}$ is a ${\displaystyle C^{k}}$ bump function. Every real-valued test function on ${\displaystyle U}$ is also a complex-valued test function on ${\displaystyle U.}$

For all ${\displaystyle j,k\in \{0,1,2,\ldots ,\infty \}}$ and any compact subsets ${\displaystyle K}$ and ${\displaystyle L}$ of ${\displaystyle U}$, we have: $${\displaystyle {\begin{aligned}C^{k}(K)&\subseteq C_{c}^{k}(U)\subseteq C^{k}(U)\\C^{k}(K)&\subseteq C^{k}(L)&&{\text{if }}K\subseteq L\\C^{k}(K)&\subseteq C^{j}(K)&&{\text{if }}j\leq k\\C_{c}^{k}(U)&\subseteq C_{c}^{j}(U)&&{\text{if }}j\leq k\\C^{k}(U)&\subseteq C^{j}(U)&&{\text{if }}j\leq k\\\end{aligned}}}$$

Distributions on U are continuous linear functionals on ${\displaystyle C_{c}^{\infty }(U)}$ when this vector space is endowed with a particular topology called the canonical LF-topology. The following proposition states two necessary and sufficient conditions for the continuity of a linear function on ${\displaystyle C_{c}^{\infty }(U)}$ that are often straightforward to verify.

Proposition: A linear functional T on ${\displaystyle C_{c}^{\infty }(U)}$ is continuous, and therefore a distribution, if and only if any of the following equivalent conditions is satisfied:

1. For every compact subset ${\displaystyle K\subseteq U}$ there exist constants ${\displaystyle C>0}$ and ${\displaystyle N\in \mathbb {N} }$ (dependent on ${\displaystyle K}$) such that for all ${\displaystyle f\in C_{c}^{\infty }(U)}$ with support contained in ${\displaystyle K}$, $${\displaystyle |T(f)|\leq C\sup\{|\partial ^{\alpha }f(x)|:x\in U,|\alpha |\leq N\}.}$$
2. For every compact subset ${\displaystyle K\subseteq U}$ and every sequence ${\displaystyle \{f_{i}\}_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{\infty }(U)}$ whose supports are contained in ${\displaystyle K}$, if ${\displaystyle \{\partial ^{\alpha }f_{i}\}_{i=1}^{\infty }}$ converges uniformly to zero on ${\displaystyle U}$ for every multi-index ${\displaystyle \alpha }$, then ${\displaystyle T(f_{i})\to 0.}$

Topology on C(U)

We now introduce the seminorms that will define the topology on ${\displaystyle C^{k}(U).}$ Different authors sometimes use different families of seminorms so we list the most common families below. However, the resulting topology is the same no matter which family is used.

$${\displaystyle {\begin{alignedat}{4}{\text{ (1) }}\ &s_{p,K}(f)&&:=\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\\{\text{ (2) }}\ &q_{i,K}(f)&&:=\sup _{|p|\leq i}\left(\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\right)=\sup _{|p|\leq i}\left(s_{p,K}(f)\right)\\{\text{ (3) }}\ &r_{i,K}(f)&&:=\sup _{\stackrel {|p|\leq i}{x_{0}\in K}}\left|\partial ^{p}f(x_{0})\right|\\{\text{ (4) }}\ &t_{i,K}(f)&&:=\sup _{x_{0}\in K}\left(\sum _{|p|\leq i}\left|\partial ^{p}f(x_{0})\right|\right)\end{alignedat}}}$$

All of the functions above are non-negative ${\displaystyle \mathbb {R} }$-valued seminorms on ${\displaystyle C^{k}(U).}$ As explained in this article, every set of seminorms on a vector space induces a locally convex vector topology.

Each of the following sets of seminorms $${\displaystyle {\begin{alignedat}{4}A~:=\quad &\{q_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\B~:=\quad &\{r_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\C~:=\quad &\{t_{i,K}&&:\;K{\text{ compact and }}\;&&i\in \mathbb {N} {\text{ satisfies }}\;&&0\leq i\leq k\}\\D~:=\quad &\{s_{p,K}&&:\;K{\text{ compact and }}\;&&p\in \mathbb {N} ^{n}{\text{ satisfies }}\;&&|p|\leq k\}\end{alignedat}}}$$ generate the same locally convex vector topology on ${\displaystyle C^{k}(U)}$ (so for example, the topology generated by the seminorms in ${\displaystyle A}$ is equal to the topology generated by those in ${\displaystyle C}$).

With this topology, ${\displaystyle C^{k}(U)}$ becomes a locally convex Fréchet space that is not normable. Every element of ${\displaystyle A\cup B\cup C\cup D}$ is a continuous seminorm on ${\displaystyle C^{k}(U).}$ Under this topology, a net ${\displaystyle (f_{i})_{i\in I}}$ in ${\displaystyle C^{k}(U)}$ converges to ${\displaystyle f\in C^{k}(U)}$ if and only if for every multi-index ${\displaystyle p}$ with ${\displaystyle |p|<k+1}$ and every compact ${\displaystyle K,}$ the net of partial derivatives ${\displaystyle \left(\partial ^{p}f_{i}\right)_{i\in I}}$ converges uniformly to ${\displaystyle \partial ^{p}f}$ on ${\displaystyle K.}$ For any ${\displaystyle k\in \{0,1,2,\ldots ,\infty \},}$ any (von Neumann) bounded subset of ${\displaystyle C^{k+1}(U)}$ is a relatively compact subset of ${\displaystyle C^{k}(U).}$ In particular, a subset of ${\displaystyle C^{\infty }(U)}$ is bounded if and only if it is bounded in ${\displaystyle C^{i}(U)}$ for all ${\displaystyle i\in \mathbb {N} .}$ The space ${\displaystyle C^{k}(U)}$ is a Montel space if and only if ${\displaystyle k=\infty .}$

A subset ${\displaystyle W}$ of ${\displaystyle C^{\infty }(U)}$ is open in this topology if and only if there exists ${\displaystyle i\in \mathbb {N} }$ such that ${\displaystyle W}$ is open when ${\displaystyle C^{\infty }(U)}$ is endowed with the subspace topology induced on it by ${\displaystyle C^{i}(U).}$

Topology on C(K)

As before, fix ${\displaystyle k\in \{0,1,2,\ldots ,\infty \}.}$ Recall that if ${\displaystyle K}$ is any compact subset of ${\displaystyle U}$ then ${\displaystyle C^{k}(K)\subseteq C^{k}(U).}$

If ${\displaystyle k}$ is finite then ${\displaystyle C^{k}(K)}$ is a Banach space with a topology that can be defined by the norm $${\displaystyle r_{K}(f):=\sup _{|p|<k}\left(\sup _{x_{0}\in K}\left|\partial ^{p}f(x_{0})\right|\right).}$$ And when ${\displaystyle k=2,}$ then ${\displaystyle C^{k}(K)}$ is even a Hilbert space.

Trivial extensions and independence of C(K)'s topology from U

Suppose ${\displaystyle U}$ is an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle K\subseteq U}$ is a compact subset. By definition, elements of ${\displaystyle C^{k}(K)}$ are functions with domain ${\displaystyle U}$ (in symbols, ${\displaystyle C^{k}(K)\subseteq C^{k}(U)}$), so the space ${\displaystyle C^{k}(K)}$ and its topology depend on ${\displaystyle U;}$ to make this dependence on the open set ${\displaystyle U}$ clear, temporarily denote ${\displaystyle C^{k}(K)}$ by ${\displaystyle C^{k}(K;U).}$ Importantly, changing the set ${\displaystyle U}$ to a different open subset ${\displaystyle U'}$ (with ${\displaystyle K\subseteq U'}$) will change the set ${\displaystyle C^{k}(K)}$ from ${\displaystyle C^{k}(K;U)}$ to ${\displaystyle C^{k}(K;U'),}$ so that elements of ${\displaystyle C^{k}(K)}$ will be functions with domain ${\displaystyle U'}$ instead of ${\displaystyle U.}$ Despite ${\displaystyle C^{k}(K)}$ depending on the open set (${\displaystyle U{\text{ or }}U'}$), the standard notation for ${\displaystyle C^{k}(K)}$ makes no mention of it. This is justified because, as this subsection will now explain, the space ${\displaystyle C^{k}(K;U)}$ is canonically identified as a subspace of ${\displaystyle C^{k}(K;U')}$ (both algebraically and topologically).

It is enough to explain how to canonically identify ${\displaystyle C^{k}(K;U)}$ with ${\displaystyle C^{k}(K;U')}$ when one of ${\displaystyle U}$ and ${\displaystyle U'}$ is a subset of the other. The reason is that if ${\displaystyle V}$ and ${\displaystyle W}$ are arbitrary open subsets of ${\displaystyle \mathbb {R} ^{n}}$ containing ${\displaystyle K}$ then the open set ${\displaystyle U:=V\cap W}$ also contains ${\displaystyle K,}$ so that each of ${\displaystyle C^{k}(K;V)}$ and ${\displaystyle C^{k}(K;W)}$ is canonically identified with ${\displaystyle C^{k}(K;V\cap W)}$ and now by transitivity, ${\displaystyle C^{k}(K;V)}$ is thus identified with ${\displaystyle C^{k}(K;W).}$ So assume ${\displaystyle U\subseteq V}$ are open subsets of ${\displaystyle \mathbb {R} ^{n}}$ containing ${\displaystyle K.}$

Given ${\displaystyle f\in C_{c}^{k}(U),}$ its trivial extension to ${\displaystyle V}$ is the function ${\displaystyle F:V\to \mathbb {C} }$ defined by: $${\displaystyle F(x)={\begin{cases}f(x)&x\in U,\\0&{\text{otherwise}}.\end{cases}}}$$ This trivial extension belongs to ${\displaystyle C^{k}(V)}$ (because ${\displaystyle f\in C_{c}^{k}(U)}$ has compact support) and it will be denoted by ${\displaystyle I(f)}$ (that is, ${\displaystyle I(f):=F}$). The assignment ${\displaystyle f\mapsto I(f)}$ thus induces a map ${\displaystyle I:C_{c}^{k}(U)\to C^{k}(V)}$ that sends a function in ${\displaystyle C_{c}^{k}(U)}$ to its trivial extension on ${\displaystyle V.}$ This map is a linear injection and for every compact subset ${\displaystyle K\subseteq U}$ (where ${\displaystyle K}$ is also a compact subset of ${\displaystyle V}$ since ${\displaystyle K\subseteq U\subseteq V}$), $${\displaystyle {\begin{alignedat}{4}I\left(C^{k}(K;U)\right)&~=~C^{k}(K;V)\qquad {\text{ and thus }}\\I\left(C_{c}^{k}(U)\right)&~\subseteq ~C_{c}^{k}(V).\end{alignedat}}}$$ If ${\displaystyle I}$ is restricted to ${\displaystyle C^{k}(K;U)}$ then the following induced linear map is a homeomorphism (linear homeomorphisms are called TVS-isomorphisms): $${\displaystyle {\begin{alignedat}{4}\,&C^{k}(K;U)&&\to \,&&C^{k}(K;V)\\&f&&\mapsto \,&&I(f)\\\end{alignedat}}}$$ and thus the next map is a topological embedding: $${\displaystyle {\begin{alignedat}{4}\,&C^{k}(K;U)&&\to \,&&C^{k}(V)\\&f&&\mapsto \,&&I(f).\\\end{alignedat}}}$$ Using the injection $${\displaystyle I:C_{c}^{k}(U)\to C^{k}(V)}$$ the vector space ${\displaystyle C_{c}^{k}(U)}$ is canonically identified with its image in ${\displaystyle C_{c}^{k}(V)\subseteq C^{k}(V).}$ Because ${\displaystyle C^{k}(K;U)\subseteq C_{c}^{k}(U),}$ through this identification, ${\displaystyle C^{k}(K;U)}$ can also be considered as a subset of ${\displaystyle C^{k}(V).}$ Thus the topology on ${\displaystyle C^{k}(K;U)}$ is independent of the open subset ${\displaystyle U}$ of ${\displaystyle \mathbb {R} ^{n}}$ that contains ${\displaystyle K,}$ which justifies the practice of writing ${\displaystyle C^{k}(K)}$ instead of ${\displaystyle C^{k}(K;U).}$

Canonical LF topology

Recall that ${\displaystyle C_{c}^{k}(U)}$ denotes all functions in ${\displaystyle C^{k}(U)}$ that have compact support in ${\displaystyle U,}$ where note that ${\displaystyle C_{c}^{k}(U)}$ is the union of all ${\displaystyle C^{k}(K)}$ as ${\displaystyle K}$ ranges over all compact subsets of ${\displaystyle U.}$ Moreover, for each ${\displaystyle k,\,C_{c}^{k}(U)}$ is a dense subset of ${\displaystyle C^{k}(U).}$ The special case when ${\displaystyle k=\infty }$ gives us the space of test functions.

space of test functions on ${\displaystyle U}$

The canonical LF-topology is not metrizable and importantly, it is strictly finer than the subspace topology that ${\displaystyle C^{\infty }(U)}$ induces on ${\displaystyle C_{c}^{\infty }(U).}$ However, the canonical LF-topology does make ${\displaystyle C_{c}^{\infty }(U)}$ into a complete reflexive nuclear Montel bornological barrelled Mackey space; the same is true of its strong dual space (that is, the space of all distributions with its usual topology). The canonical LF-topology can be defined in various ways.

Distributions

As discussed earlier, continuous linear functionals on a ${\displaystyle C_{c}^{\infty }(U)}$ are known as distributions on ${\displaystyle U.}$ Other equivalent definitions are described below.

distribution on ${\displaystyle U}$

There is a canonical duality pairing between a distribution ${\displaystyle T}$ on ${\displaystyle U}$ and a test function ${\displaystyle f\in C_{c}^{\infty }(U),}$ which is denoted using angle brackets by $${\displaystyle {\begin{cases}{\mathcal {D}}'(U)\times C_{c}^{\infty }(U)\to \mathbb {R} \\(T,f)\mapsto \langle T,f\rangle :=T(f)\end{cases}}}$$

One interprets this notation as the distribution ${\displaystyle T}$ acting on the test function ${\displaystyle f}$ to give a scalar, or symmetrically as the test function ${\displaystyle f}$ acting on the distribution ${\displaystyle T.}$

Characterizations of distributions

Proposition. If ${\displaystyle T}$ is a linear functional on ${\displaystyle C_{c}^{\infty }(U)}$ then the following are equivalent:

1. T is a distribution;
2. T is continuous;
3. T is continuous at the origin;
4. T is uniformly continuous;
5. T is a bounded operator;
6. T is sequentially continuous;
   • explicitly, for every sequence ${\displaystyle \left(f_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{\infty }(U)}$ that converges in ${\displaystyle C_{c}^{\infty }(U)}$ to some ${\displaystyle f\in C_{c}^{\infty }(U),}$ ${\textstyle \lim _{i\to \infty }T\left(f_{i}\right)=T(f);}$
7. T is sequentially continuous at the origin; in other words, T maps null sequences to null sequences;
   • explicitly, for every sequence ${\displaystyle \left(f_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{\infty }(U)}$ that converges in ${\displaystyle C_{c}^{\infty }(U)}$ to the origin (such a sequence is called a null sequence), ${\textstyle \lim _{i\to \infty }T\left(f_{i}\right)=0;}$
   • a null sequence is by definition any sequence that converges to the origin;
8. T maps null sequences to bounded subsets;
   • explicitly, for every sequence ${\displaystyle \left(f_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{\infty }(U)}$ that converges in ${\displaystyle C_{c}^{\infty }(U)}$ to the origin, the sequence ${\displaystyle \left(T\left(f_{i}\right)\right)_{i=1}^{\infty }}$ is bounded;
9. T maps Mackey convergent null sequences to bounded subsets;
   • explicitly, for every Mackey convergent null sequence ${\displaystyle \left(f_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle C_{c}^{\infty }(U),}$ the sequence ${\displaystyle \left(T\left(f_{i}\right)\right)_{i=1}^{\infty }}$ is bounded;
   • a sequence ${\displaystyle f_{\bullet }=\left(f_{i}\right)_{i=1}^{\infty }}$ is said to be Mackey convergent to the origin if there exists a divergent sequence ${\displaystyle r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }\to \infty }$ of positive real numbers such that the sequence ${\displaystyle \left(r_{i}f_{i}\right)_{i=1}^{\infty }}$ is bounded; every sequence that is Mackey convergent to the origin necessarily converges to the origin (in the usual sense);
10. The kernel of T is a closed subspace of ${\displaystyle C_{c}^{\infty }(U);}$
11. The graph of T is closed;
12. There exists a continuous seminorm ${\displaystyle g}$ on ${\displaystyle C_{c}^{\infty }(U)}$ such that ${\displaystyle |T|\leq g;}$
13. There exists a constant ${\displaystyle C>0}$ and a finite subset ${\displaystyle \{g_{1},\ldots ,g_{m}\}\subseteq {\mathcal {P}}}$ (where ${\displaystyle {\mathcal {P}}}$ is any collection of continuous seminorms that defines the canonical LF topology on ${\displaystyle C_{c}^{\infty }(U)}$) such that ${\displaystyle |T|\leq C(g_{1}+\cdots +g_{m});}$
14. For every compact subset ${\displaystyle K\subseteq U}$ there exist constants ${\displaystyle C>0}$ and ${\displaystyle N\in \mathbb {N} }$ such that for all ${\displaystyle f\in C^{\infty }(K),}$ $${\displaystyle |T(f)|\leq C\sup\{|\partial ^{\alpha }f(x)|:x\in U,|\alpha |\leq N\};}$$
15. For every compact subset ${\displaystyle K\subseteq U}$ there exist constants ${\displaystyle C_{K}>0}$ and ${\displaystyle N_{K}\in \mathbb {N} }$ such that for all ${\displaystyle f\in C_{c}^{\infty }(U)}$ with support contained in ${\displaystyle K,}$ $${\displaystyle |T(f)|\leq C_{K}\sup\{|\partial ^{\alpha }f(x)|:x\in K,|\alpha |\leq N_{K}\};}$$
16. For any compact subset ${\displaystyle K\subseteq U}$ and any sequence ${\displaystyle \{f_{i}\}_{i=1}^{\infty }}$ in ${\displaystyle C^{\infty }(K),}$ if ${\displaystyle \{\partial ^{p}f_{i}\}_{i=1}^{\infty }}$ converges uniformly to zero for all multi-indices ${\displaystyle p,}$ then ${\displaystyle T(f_{i})\to 0;}$

Topology on the space of distributions and its relation to the weak-* topology

The set of all distributions on ${\displaystyle U}$ is the continuous dual space of ${\displaystyle C_{c}^{\infty }(U),}$ which when endowed with the strong dual topology is denoted by ${\displaystyle {\mathcal {D}}'(U).}$ Importantly, unless indicated otherwise, the topology on ${\displaystyle {\mathcal {D}}'(U)}$ is the strong dual topology; if the topology is instead the weak-* topology then this will be indicated. Neither topology is metrizable although unlike the weak-* topology, the strong dual topology makes ${\displaystyle {\mathcal {D}}'(U)}$ into a complete nuclear space, to name just a few of its desirable properties.

Neither ${\displaystyle C_{c}^{\infty }(U)}$ nor its strong dual ${\displaystyle {\mathcal {D}}'(U)}$ is a sequential space and so neither of their topologies can be fully described by sequences (in other words, defining only what sequences converge in these spaces is not enough to fully/correctly define their topologies). However, a sequence in ${\displaystyle {\mathcal {D}}'(U)}$ converges in the strong dual topology if and only if it converges in the weak-* topology (this leads many authors to use pointwise convergence to define the convergence of a sequence of distributions; this is fine for sequences but this is not guaranteed to extend to the convergence of nets of distributions because a net may converge pointwise but fail to converge in the strong dual topology). More information about the topology that ${\displaystyle {\mathcal {D}}'(U)}$ is endowed with can be found in the article on spaces of test functions and distributions and the articles on polar topologies and dual systems.

A linear map from ${\displaystyle {\mathcal {D}}'(U)}$ into another locally convex topological vector space (such as any normed space) is continuous if and only if it is sequentially continuous at the origin. However, this is no longer guaranteed if the map is not linear or for maps valued in more general topological spaces (for example, that are not also locally convex topological vector spaces). The same is true of maps from ${\displaystyle C_{c}^{\infty }(U)}$ (more generally, this is true of maps from any locally convex bornological space).

Localization of distributions

There is no way to define the value of a distribution in ${\displaystyle {\mathcal {D}}'(U)}$ at a particular point of U. However, as is the case with functions, distributions on U restrict to give distributions on open subsets of U. Furthermore, distributions are locally determined in the sense that a distribution on all of U can be assembled from a distribution on an open cover of U satisfying some compatibility conditions on the overlaps. Such a structure is known as a sheaf.

Extensions and restrictions to an open subset

Let ${\displaystyle V\subseteq U}$ be open subsets of ${\displaystyle \mathbb {R} ^{n}.}$ Every function ${\displaystyle f\in {\mathcal {D}}(V)}$ can be extended by zero from its domain V to a function on U by setting it equal to ${\displaystyle 0}$ on the complement ${\displaystyle U\setminus V.}$ This extension is a smooth compactly supported function called the trivial extension of ${\displaystyle f}$ to ${\displaystyle U}$ and it will be denoted by ${\displaystyle E_{VU}(f).}$ This assignment ${\displaystyle f\mapsto E_{VU}(f)}$ defines the trivial extension operator ${\displaystyle E_{VU}:{\mathcal {D}}(V)\to {\mathcal {D}}(U),}$ which is a continuous injective linear map. It is used to canonically identify ${\displaystyle {\mathcal {D}}(V)}$ as a vector subspace of ${\displaystyle {\mathcal {D}}(U)}$ (although not as a topological subspace). Its transpose (explained here) $${\displaystyle \rho _{VU}:={}^{t}E_{VU}:{\mathcal {D}}'(U)\to {\mathcal {D}}'(V),}$$ is called the restriction to ${\displaystyle V}$ of distributions in ${\displaystyle U}$ and as the name suggests, the image ${\displaystyle \rho _{VU}(T)}$ of a distribution ${\displaystyle T\in {\mathcal {D}}'(U)}$ under this map is a distribution on ${\displaystyle V}$ called the restriction of ${\displaystyle T}$ to ${\displaystyle V.}$ The defining condition of the restriction ${\displaystyle \rho _{VU}(T)}$ is: $${\displaystyle \langle \rho _{VU}T,\phi \rangle =\langle T,E_{VU}\phi \rangle \quad {\text{ for all }}\phi \in {\mathcal {D}}(V).}$$ If ${\displaystyle V\neq U}$ then the (continuous injective linear) trivial extension map ${\displaystyle E_{VU}:{\mathcal {D}}(V)\to {\mathcal {D}}(U)}$ is not a topological embedding (in other words, if this linear injection was used to identify ${\displaystyle {\mathcal {D}}(V)}$ as a subset of ${\displaystyle {\mathcal {D}}(U)}$ then ${\displaystyle {\mathcal {D}}(V)}$'s topology would strictly finer than the subspace topology that ${\displaystyle {\mathcal {D}}(U)}$ induces on it; importantly, it would not be a topological subspace since that requires equality of topologies) and its range is also not dense in its codomain ${\displaystyle {\mathcal {D}}(U).}$ Consequently if ${\displaystyle V\neq U}$ then the restriction mapping is neither injective nor surjective. A distribution ${\displaystyle S\in {\mathcal {D}}'(V)}$ is said to be extendible to U if it belongs to the range of the transpose of ${\displaystyle E_{VU}}$ and it is called extendible if it is extendable to ${\displaystyle \mathbb {R} ^{n}.}$

Unless ${\displaystyle U=V,}$ the restriction to V is neither injective nor surjective. Lack of surjectivity follows since distributions can blow up towards the boundary of V. For instance, if ${\displaystyle U=\mathbb {R} }$ and ${\displaystyle V=(0,2),}$ then the distribution $${\displaystyle T(x)=\sum _{n=1}^{\infty }n\,\delta \left(x-{\frac {1}{n}}\right)}$$ is in ${\displaystyle {\mathcal {D}}'(V)}$ but admits no extension to ${\displaystyle {\mathcal {D}}'(U).}$

Gluing and distributions that vanish in a set

Theorem — Let ${\displaystyle (U_{i})_{i\in I}}$ be a collection of open subsets of ${\displaystyle \mathbb {R} ^{n}.}$ For each ${\displaystyle i\in I,}$ let ${\displaystyle T_{i}\in {\mathcal {D}}'(U_{i})}$ and suppose that for all ${\displaystyle i,j\in I,}$ the restriction of ${\displaystyle T_{i}}$ to ${\displaystyle U_{i}\cap U_{j}}$ is equal to the restriction of ${\displaystyle T_{j}}$ to ${\displaystyle U_{i}\cap U_{j}}$ (note that both restrictions are elements of ${\displaystyle {\mathcal {D}}'(U_{i}\cap U_{j})}$). Then there exists a unique ${\textstyle T\in {\mathcal {D}}'(\bigcup _{i\in I}U_{i})}$ such that for all ${\displaystyle i\in I,}$ the restriction of T to ${\displaystyle U_{i}}$ is equal to ${\displaystyle T_{i}.}$

Let V be an open subset of U. ${\displaystyle T\in {\mathcal {D}}'(U)}$ is said to vanish in V if for all ${\displaystyle f\in {\mathcal {D}}(U)}$ such that ${\displaystyle \operatorname {supp} (f)\subseteq V}$ we have ${\displaystyle Tf=0.}$ T vanishes in V if and only if the restriction of T to V is equal to 0, or equivalently, if and only if T lies in the kernel of the restriction map ${\displaystyle \rho _{VU}.}$

Corollary — Let ${\displaystyle (U_{i})_{i\in I}}$ be a collection of open subsets of ${\displaystyle \mathbb {R} ^{n}}$ and let ${\textstyle T\in {\mathcal {D}}'(\bigcup _{i\in I}U_{i}).}$ ${\displaystyle T=0}$ if and only if for each ${\displaystyle i\in I,}$ the restriction of T to ${\displaystyle U_{i}}$ is equal to 0.

Corollary — The union of all open subsets of U in which a distribution T vanishes is an open subset of U in which T vanishes.

Support of a distribution

This last corollary implies that for every distribution T on U, there exists a unique largest subset V of U such that T vanishes in V (and does not vanish in any open subset of U that is not contained in V); the complement in U of this unique largest open subset is called the support of T. Thus $${\displaystyle \operatorname {supp} (T)=U\setminus \bigcup \{V\mid \rho _{VU}T=0\}.}$$

If ${\displaystyle f}$ is a locally integrable function on U and if ${\displaystyle D_{f}}$ is its associated distribution, then the support of ${\displaystyle D_{f}}$ is the smallest closed subset of U in the complement of which ${\displaystyle f}$ is almost everywhere equal to 0. If ${\displaystyle f}$ is continuous, then the support of ${\displaystyle D_{f}}$ is equal to the closure of the set of points in U at which ${\displaystyle f}$ does not vanish. The support of the distribution associated with the Dirac measure at a point ${\displaystyle x_{0}}$ is the set ${\displaystyle \{x_{0}\}.}$ If the support of a test function ${\displaystyle f}$ does not intersect the support of a distribution T then ${\displaystyle Tf=0.}$ A distribution T is 0 if and only if its support is empty. If ${\displaystyle f\in C^{\infty }(U)}$ is identically 1 on some open set containing the support of a distribution T then ${\displaystyle fT=T.}$ If the support of a distribution T is compact then it has finite order and there is a constant ${\displaystyle C}$ and a non-negative integer ${\displaystyle N}$ such that: $${\displaystyle |T\phi |\leq C\|\phi \|_{N}:=C\sup \left\{\left|\partial ^{\alpha }\phi (x)\right|:x\in U,|\alpha |\leq N\right\}\quad {\text{ for all }}\phi \in {\mathcal {D}}(U).}$$

If T has compact support, then it has a unique extension to a continuous linear functional ${\displaystyle {\widehat {T}}}$ on ${\displaystyle C^{\infty }(U)}$; this function can be defined by ${\displaystyle {\widehat {T}}(f):=T(\psi f),}$ where ${\displaystyle \psi \in {\mathcal {D}}(U)}$ is any function that is identically 1 on an open set containing the support of T.

If ${\displaystyle S,T\in {\mathcal {D}}'(U)}$ and ${\displaystyle \lambda \neq 0}$ then ${\displaystyle \operatorname {supp} (S+T)\subseteq \operatorname {supp} (S)\cup \operatorname {supp} (T)}$ and ${\displaystyle \operatorname {supp} (\lambda T)=\operatorname {supp} (T).}$ Thus, distributions with support in a given subset ${\displaystyle A\subseteq U}$ form a vector subspace of ${\displaystyle {\mathcal {D}}'(U).}$ Furthermore, if ${\displaystyle P}$ is a differential operator in U, then for all distributions T on U and all ${\displaystyle f\in C^{\infty }(U)}$ we have ${\displaystyle \operatorname {supp} (P(x,\partial )T)\subseteq \operatorname {supp} (T)}$ and ${\displaystyle \operatorname {supp} (fT)\subseteq \operatorname {supp} (f)\cap \operatorname {supp} (T).}$

Distributions with compact support

Support in a point set and Dirac measures

For any ${\displaystyle x\in U,}$ let ${\displaystyle \delta _{x}\in {\mathcal {D}}'(U)}$ denote the distribution induced by the Dirac measure at ${\displaystyle x.}$ For any ${\displaystyle x_{0}\in U}$ and distribution ${\displaystyle T\in {\mathcal {D}}'(U),}$ the support of T is contained in ${\displaystyle \{x_{0}\}}$ if and only if T is a finite linear combination of derivatives of the Dirac measure at ${\displaystyle x_{0}.}$ If in addition the order of T is ${\displaystyle \leq k}$ then there exist constants ${\displaystyle \alpha _{p}}$ such that: $${\displaystyle T=\sum _{|p|\leq k}\alpha _{p}\partial ^{p}\delta _{x_{0}}.}$$

Said differently, if T has support at a single point ${\displaystyle \{P\},}$ then T is in fact a finite linear combination of distributional derivatives of the ${\displaystyle \delta }$ function at P. That is, there exists an integer m and complex constants ${\displaystyle a_{\alpha }}$ such that $${\displaystyle T=\sum _{|\alpha |\leq m}a_{\alpha }\partial ^{\alpha }(\tau _{P}\delta )}$$ where ${\displaystyle \tau _{P}}$ is the translation operator.

Distribution with compact support

Theorem — Suppose T is a distribution on U with compact support K. There exists a continuous function ${\displaystyle f}$ defined on U and a multi-index p such that $${\displaystyle T=\partial ^{p}f,}$$ where the derivatives are understood in the sense of distributions. That is, for all test functions ${\displaystyle \phi }$ on U, $${\displaystyle T\phi =(-1)^{|p|}\int _{U}f(x)(\partial ^{p}\phi )(x)\,dx.}$$

Distributions of finite order with support in an open subset

Theorem — Suppose T is a distribution on U with compact support K and let V be an open subset of U containing K. Since every distribution with compact support has finite order, take N to be the order of T and define ${\displaystyle P:=\{0,1,\ldots ,N+2\}^{n}.}$ There exists a family of continuous functions ${\displaystyle (f_{p})_{p\in P}}$ defined on U with support in V such that $${\displaystyle T=\sum _{p\in P}\partial ^{p}f_{p},}$$ where the derivatives are understood in the sense of distributions. That is, for all test functions ${\displaystyle \phi }$ on U, $${\displaystyle T\phi =\sum _{p\in P}(-1)^{|p|}\int _{U}f_{p}(x)(\partial ^{p}\phi )(x)\,dx.}$$

Global structure of distributions

The formal definition of distributions exhibits them as a subspace of a very large space, namely the topological dual of ${\displaystyle {\mathcal {D}}(U)}$ (or the Schwartz space ${\displaystyle {\mathcal {S}}(\mathbb {R} ^{n})}$ for tempered distributions). It is not immediately clear from the definition how exotic a distribution might be. To answer this question, it is instructive to see distributions built up from a smaller space, namely the space of continuous functions. Roughly, any distribution is locally a (multiple) derivative of a continuous function. A precise version of this result, given below, holds for distributions of compact support, tempered distributions, and general distributions. Generally speaking, no proper subset of the space of distributions contains all continuous functions and is closed under differentiation. This says that distributions are not particularly exotic objects; they are only as complicated as necessary.

Distributions as sheaves

Theorem — Let T be a distribution on U. There exists a sequence ${\displaystyle (T_{i})_{i=1}^{\infty }}$ in ${\displaystyle {\mathcal {D}}'(U)}$ such that each T_i has compact support and every compact subset ${\displaystyle K\subseteq U}$ intersects the support of only finitely many ${\displaystyle T_{i},}$ and the sequence of partial sums ${\displaystyle (S_{j})_{j=1}^{\infty },}$ defined by ${\displaystyle S_{j}:=T_{1}+\cdots +T_{j},}$ converges in ${\displaystyle {\mathcal {D}}'(U)}$ to T; in other words we have: $${\displaystyle T=\sum _{i=1}^{\infty }T_{i}.}$$ Recall that a sequence converges in ${\displaystyle {\mathcal {D}}'(U)}$ (with its strong dual topology) if and only if it converges pointwise.

Decomposition of distributions as sums of derivatives of continuous functions

By combining the above results, one may express any distribution on U as the sum of a series of distributions with compact support, where each of these distributions can in turn be written as a finite sum of distributional derivatives of continuous functions on U. In other words, for arbitrary ${\displaystyle T\in {\mathcal {D}}'(U)}$ we can write: $${\displaystyle T=\sum _{i=1}^{\infty }\sum _{p\in P_{i}}\partial ^{p}f_{ip},}$$ where ${\displaystyle P_{1},P_{2},\ldots }$ are finite sets of multi-indices and the functions ${\displaystyle f_{ip}}$ are continuous.

Theorem — Let T be a distribution on U. For every multi-index p there exists a continuous function ${\displaystyle g_{p}}$ on U such that

1. any compact subset K of U intersects the support of only finitely many ${\displaystyle g_{p},}$ and
2. ${\displaystyle T=\sum \nolimits _{p}\partial ^{p}g_{p}.}$

Moreover, if T has finite order, then one can choose ${\displaystyle g_{p}}$ in such a way that only finitely many of them are non-zero.

Note that the infinite sum above is well-defined as a distribution. The value of T for a given ${\displaystyle f\in {\mathcal {D}}(U)}$ can be computed using the finitely many ${\displaystyle g_{\alpha }}$ that intersect the support of ${\displaystyle f.}$

Operations on distributions

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if ${\displaystyle A:{\mathcal {D}}(U)\to {\mathcal {D}}(U)}$ is a linear map that is continuous with respect to the weak topology, then it is not always possible to extend ${\displaystyle A}$ to a map ${\displaystyle A':{\mathcal {D}}'(U)\to {\mathcal {D}}'(U)}$ by classic extension theorems of topology or linear functional analysis. The “distributional” extension of the above linear continuous operator A is possible if and only if A admits a Schwartz adjoint, that is another linear continuous operator B of the same type such that ${\displaystyle \langle Af,g\rangle =\langle f,Bg\rangle }$, for every pair of test functions. In that condition, B is unique and the extension A’ is the transpose of the Schwartz adjoint B.

Preliminaries: Transpose of a linear operator

Operations on distributions and spaces of distributions are often defined using the transpose of a linear operator. This is because the transpose allows for a unified presentation of the many definitions in the theory of distributions and also because its properties are well-known in functional analysis. For instance, the well-known Hermitian adjoint of a linear operator between Hilbert spaces is just the operator's transpose (but with the Riesz representation theorem used to identify each Hilbert space with its continuous dual space). In general, the transpose of a continuous linear map ${\displaystyle A:X\to Y}$ is the linear map $${\displaystyle {}^{t}A:Y'\to X'\qquad {\text{ defined by }}\qquad {}^{t}A(y'):=y'\circ A,}$$ or equivalently, it is the unique map satisfying ${\displaystyle \langle y',A(x)\rangle =\left\langle {}^{t}A(y'),x\right\rangle }$ for all ${\displaystyle x\in X}$ and all ${\displaystyle y'\in Y'}$ (the prime symbol in ${\displaystyle y'}$ does not denote a derivative of any kind; it merely indicates that ${\displaystyle y'}$ is an element of the continuous dual space ${\displaystyle Y'}$). Since ${\displaystyle A}$ is continuous, the transpose ${\displaystyle {}^{t}A:Y'\to X'}$ is also continuous when both duals are endowed with their respective strong dual topologies; it is also continuous when both duals are endowed with their respective weak* topologies (see the articles polar topology and dual system for more details).